Interpreting Loop Gain Measurements
How to read the critical regions of a loop gain and phase measurement.
There have been many dramatic changes in power supply development over the last 20 years, but loop gain measurements remain the key to rugged and aggressive system performance. Understanding how to read a loop gain is important.
Loop Gain Measurements in Modern Control Systems
Quite a few years ago, I emerged from college to enter the world of commercial power supply design. I had studied microprocessors, optimal control theory, multi-state feedback, and I was ready to tackle some real hardware and put everything I thought I knew into practice. It was a time when current-mode control was just coming into use, and I could see that current-mode was a classic example of multi-state feedback. All we needed to do was determine the proper gains from each state, and we could do placement of the closed-loop poles wherever we wanted them - just like in college!
But there was a problem. Nobody at work knew what I was talking about. And, unlike in the problem sets in college courses, nobody could tell me where they wanted the closed-loop poles to be. They all talked in strange terms like output impedance, loop gains, audiosusceptibility, and it wasn’t clear what to do next.
I remember three things clearly from Middlebrook's famous analog electronics course - how a zero-ripple Cuk worked, a new way to solve the quadratic equation, and the need to measure loop gains in power supplies.
Then I attended Middlebrook’s famous course on analog circuit design. It was a long time ago, but there are three things I can remember clearly from that course:
First, he measured loop gains for all his power supply and other analog examples and injected into the loop using a current probe driven backwards from an oscillator. A very neat trick, all you had to do was put a loop of wire in the feedback path and clip on the current probe.
The second thing I remember was the zero-ripple Cuk converter. There was a transparency (pre-Powerpoint days!) where he rotated a picture of coupled cores and as the gap changed on the core, the ripple current flattened out to zero on the input and output. It was a great visual that really drove the point home.
And finally, he showed that the classic solution to the quadratic equation using the usual b2 – 4ac radical was numerically inaccurate, and he gave a much better solution.
I haven’t used his quadratic solution since then, nor have I designed with a coupled-inductor Cuk converter. But once I left his course I started measuring and understanding loop gains and found that they have never gone out of style for switching power supplies. Archaic though they seemed to me at the time, they are simply the best way to optimize the feedback of your power supply. Even if you are using a digital controller, an analog loop gain is simply the best way to verify that the feedback system is designed and working properly.
If you understand how to interpret loops properly, they are all you need for stability analysis. Text books talk about Nyquist plots, and characteristic equations, but in the real world we need to use the incredibly powerful tool for engineers that Mr. Bode gave to us. It is an amazing thing – with a couple of pen strokes on paper showing the loop gain and phase, we can determine the stability of systems of almost any order. What a potent engineering tool, no math needed, no calculus, only lab measurements! This was the great contribution of Bode.
The great contribution of Bode was that he enabled engineers to draw a couple of lines on a sheet of paper and declare whether a highly-complex, high-order, nonlinear system was stable or not! No match, no calculus needed. What more could we ask for?
The aerospace design world is probably the most rigorous in making complete sets of bode plots for input impedance, output impedance, audiosusceptibility, and loop gain. Outside of the aerospace world, it is less common to make this full set of measurements. Most experienced designers will make a loop gain measurement since they find that it is a very sensitive measure of just about everything in the power stage and the feedback path. If some component is the wrong value, or something is built wrong, the loop gain is very likely to show that there is a problem.
Critical Regions of a Voltage-Mode Loop Gain Measurement
When talking about loop gains, most articles refer to just the crossover frequency and the phase margin at that frequency. In reality, there is far more to a loop gain, and if you want to derive maximum benefit from making these measurements, it is important to understand where to look.
Figure 1 shows a typical loop gain for a voltage-mode power supply. The plot of Figure 1 starts at 10 Hz. This is always recommended, regardless of your power system switching frequency. It is very common to have substantial noise in the first decade of measurement (audio people are painfully aware of this, regarding hum) and you must be able to verify that you have high gain in the low frequency regions to reject line and other low-frequency noise. This area is shown shaded in blue in Figure 1. The AP300 frequency response analyzer [2,3] is capable of measuring gains in excess of 90 dB in the presence of high noise, and this is crucial for resolving high performance systems properly.
Figure 1: Voltage-Mode Loop Gain and Phase, Showing Key Measurement Regions.
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